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A054493
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A Pellian-related recursive sequence.
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3
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1, 7, 36, 175, 841, 4032, 19321, 92575, 443556, 2125207, 10182481, 48787200, 233753521, 1119980407, 5366148516, 25710762175, 123187662361, 590227549632, 2827950085801, 13549522879375, 64919664311076, 311048798676007
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| This is the r=7 member in the r-family of sequences S_r(n+1) defined in A092184 where more information can be found.
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REFERENCES
| I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pps. 181-193.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pps. 122-125, 194-196.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
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LINKS
| R. Stephan, Boring proof of a nonlinearity
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)=5a(n-1)-a(n-2)+2, a(0)=1, a(1)=7.
a(n) = 1/3*{-2+[(5+sqrt(21))/2]^n+[(5-sqrt(21))/2]^n}. - R. Stephan, Apr 14 2004
G.f.: (1+x)/((1-x)*(1-5*x+x^2))=(1+x)/(1-6*x+6*x^2-x^3). From the R. Stephan link.
a(n)= 6*a(n-1)-6*a(n-2)+a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=7.
a(n)=2*T(n, 5/2)-2, with twice the Chebyshev's polynomials of the first kind, 2*T(n, x=5/2)=A003501(n).
a(n)= b(n) + b(n-1), n>=1, with b(n):=A089817(n) the partial sums of S(n, 5)= U(n, 5/2)=A004254(n+1), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind.
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EXAMPLE
| A004254 = sqrt{21*(A054493)^2+28*(A054493)}/7.
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CROSSREFS
| Cf. A004254.
Sequence in context: A102053 A058681 A110310 * A037538 A037482 A147546
Adjacent sequences: A054490 A054491 A054492 * A054494 A054495 A054496
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KEYWORD
| easy,nonn
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AUTHOR
| Barry E. Williams, May 06 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 10 2000
Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 10 2004
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